Friday, 28 October 2011

Is Mathematics Invented or Discovered: Thinking about the Mystery of the Primes

Prime numbers are what is left when you have taken all the patterns away. I think prime numbers are like life. They are very logical but you could never work out the rules, even if you spent all your time thinking about them.
I was recently given a book,  The Mystery of the the Prime Numbers by Matt Watkins. It is an eccentric and addictive read. I’ve become so obsessed by the primes that I’m also reading Marcus du Sautoy’s Music of the Primes at the same time.
If you are not a mathematician or interested in mathematics, you might well be asking yourself “What is so interesting about prime numbers anyway? Aren’t they just the numbers we learned about in school through the rote definition: ‘A prime is a number whose only factors are 1 and itself. That means there is no whole number that evenly divides the prime number.’” Yawn!
Yes, teachers of mathematics do often tend to limit rather than enhance our sense of curiosity. However, if you think that the primes can be limited to this stale definition, you owe yourself an exploration into  the strange, amazing landscape of the primes.  For example, there is no largest prime number, and we can never draw up a complete list of primes. Why? Well, according to the Fundamental Theorem of  Arithmetic a counting number must either be prime or split up into primes-there’s no third option. Think about it for a minute, as Euclid did many centuries ago. If there were only a finite number of primes, as he pointed out, then multiplying them all together and adding 1, would produce a number which was not divisible by an prime at all, and that’s impossible. Yes, there is an infinitude of primes.
Also fascinating is the fact that the primes refuse to conform to any pattern and for that reason they have uses in the real word for encryption and information security.  Strangely enough, as Riemann realized, they can be turned into wave functions. These are, of course, the patterns of music, of quantum waves and much more. As du Sautoy points out in his book that for  the physicist Michael Berry these waves are “not just abstract music, but can be translated into physical[if cacophonous] sounds] that anyone can listen to.”
If you’re deeply interested in fleshing this out, you might be interested in some specific work, not covered in either book , of  Fields Metalist, Terrance Tao who has described one approach to proving the prime number theorem in poetic terms–in listening to the “music” of the primes. We start with a “sound wave” that is “noisy” at the prime numbers and silent at other numbers; this is the   Von Mangoldt function. Then we analyze its notes or frequencies by subjecting it to a process akin to the Fourier transformation; this is the Merlin transformation. Then we prove, and this is the hard part, that certain “notes” cannot occur in this music. This exclusion of certain notes leads to the statement of the prime number theorem. According to Tao, this proof yields much deeper insights into the distribution of the primes than the “elementary” proofs.
I’ll likely write more on this. I can’t stop thinking about it.

1. You do not need the fundamental theorem of arithmetic to prove there are an infinite number of primes, but in any case, one would think that this simple fact about primes would be one of the very first things "teachers of mathematics" would mention about them.

And primes are not used for encryption because "they refuse to conform to any pattern". What "refusing to conform to any pattern" means is hopelessly vague. They are used for encryption because the patterns they conform to makes it easy to find large prime numbers, but no one knows in general how to efficiently factor a number which is the product of two large primes (and probably it is not possible to do efficiently except on a quantum computer).

1. Thanks...I studied the history mathematics at university, but I'm not extremely proficient in this are of maths, so I appreciate your corrections and qualifications.